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Concentration profiles in drying cylindrical filaments

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Abstract

We analyze theoretically the drying of cylindrical filaments. For modelling the mass transfer on the gas side of the liquid-gas interface of the shrinking circular cylindrical filament, we apply the model of Abramzon and Sirignano, which was originally developed for spherical geometry. As a consequence of mass transfer at constant Sherwood number, we obtain a d2-law for the shrinkage of the cylinder as in the case of the spherical geometry, which expresses that the cross-sectional area of the cylinder shrinks at a constant rate with time. For this situation, the diffusion equation for the liquid phase mixture components becomes separable upon transformation into similarity coordinates and is solved analytically to obtain the concentration profiles inside the filament as functions of time. The dependency of the profiles on the radial coordinate is determined by a series of Kummer’s functions. Applying this result, we study the evolution of the concentration profiles in the liquid phase with time as dependent on a parameter given as the ratio of rate of shrinkage of the cross-sectional area of the cylinder to liquid-phase diffusion coefficient, which was identified as relevant for the shape of the concentration profiles formed in the liquid during the drying process. As an example, we present computed results for the constant evaporation rate regime in the dry-spinning process of a polyvinyl-alcohol (PVA)-water system. Comparison of our analytical results with full numerical solutions of the diffusion equation from the literature, achieved with concentration-dependent diffusion coefficient, reveals very good agreement.

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Abbreviations

a :

time-dependent filament radius (m)

a 0 :

initial filament radius (m)

a f :

gas film radius (m)

a m :

mean radius of the filament in the constant rate period (m)

\(\tilde{a},\;\tilde{b}\) :

arguments of Kummer’s function (–)

B M :

Spalding mass transfer number (–)

C j :

expansion coefficients (–)

D :

binary diffusion coefficient in the liquid (m2/s)

\(\bar{D}_{\rm g}\) :

average binary diffusion coefficient in the gas film (m2/s)

F M :

correction factor (–)

J m :

total mass flow rate through the gas film (kg/s)

M :

confluent hypergeometric function of the first kind (Kummer’s function) (–)

\(\dot{m}\) :

evaporation mass flux (kg/m2s)

\(\dot{m}_{\rm i}\) :

evaporation rate per unit axial length of mixture component i (kg/m s)

\(\dot{m}_{{\rm evap}}\) :

evaporation rate per unit axial length (kg/m s)

n :

exponent (–)

Nu 0 :

Nusselt number at small rates of mass transfer (–)

Nu * :

modified Nusselt number (–)

\(\dot{q}\) :

heat flux in the gas film (W/m2)

Pr :

Prandtl number (–)

r :

radial coordinate (m)

Sc :

Schmidt number (–)

Sh 0 :

Sherwood number at small rates of mass transfer (–)

Sh * :

modified Sherwood number (–)

T a :

air temperature (K)

T s :

temperature at the filament surface (K)

T wb :

wet bulb temperature (K)

T ∞ :

ambient gas temperature (K)

t :

time (s)

t l :

longest possible lifetime (s)

u p :

velocity of the parallel air flow (m/s)

u w :

velocity of the moving fibre (m/s)

v r :

radial velocity of the mixture in the gas film (m/s)

V l :

volume of the filament liquid (m3)

Y i :

mass fraction of the mixture component i (–)

Y m :

mass fraction of the mixture component m in the gas film (–)

\(\bar{Y}_{20}\) :

initial mean mass fraction of the solute (–)

z :

length coordinate (m)

z F :

distance along the constant rate section (m)

α:

shrinkage rate of the filament (m2/s)

αf :

heat transfer coefficient on the gas side of the interface (W/m2K)

δfm :

thickness of the gas film (m)

λf :

heat conductivity in the gas film (W/mK)

λj :

eigenvalue (–)

ξ:

non-dimensional radial coordinate (–)

ρl :

liquid density of the solvent (kg/m3)

\(\bar{\rho}_{\rm g}\) :

mean density in the gas film (kg/m3)

ρs :

solid density of the solute (kg/m3)

τ:

non-dimensional time (–)

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Acknowledgements

The present work emerged from research projects in cooperation with the Austrian Industry. The authors gratefully acknowledge financial support from MAG Maschinen und Apparatebau AG (Deutschlandsberg, Austria) and the Austrian Research Promotion Agency (FFG) under contract numbers 809.184 and 814.748.

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Authors and Affiliations

  1. Institute of Fluid Mechanics and Heat Transfer, Graz University of Technology, Inffeldgasse 25/F, 8010, Graz, Austria

    Klaus Czaputa, Günter Brenn & Walter Meile

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  1. Klaus Czaputa
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  2. Günter Brenn
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  3. Walter Meile
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Correspondence to Günter Brenn.

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Czaputa, K., Brenn, G. & Meile, W. Concentration profiles in drying cylindrical filaments. Heat Mass Transfer 45, 227–238 (2008). https://doi.org/10.1007/s00231-008-0413-5

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